The invention relates to an electrically controlled unit fuel injector the principles and operation of which will be explained in detail hereinafter. In a unit fuel injector of this kind for Diesel engines, which is known from U.S. Pat. No. 4,392,612 and is built directly into the cylinder head of the associated internal combustion engine includes in a common housing both the mechanically driven piston injection pump and the associated injection nozzle, the fuel injection quantity which is positively displaced out of the pump work chamber to the injection nozzle during the supply stroke of the pump piston is determined by the ON time of an electromagnetically actuated overflow valve that is open when no current is flowing through it. The overflow valve is inserted into an overflow conduit joining the pump work chamber to a low-pressure chamber. The first section of the overflow conduit, which communicates constantly with the pump work chamber, is then subject to the full injection pressure whenever the overflow valve blocks the connection between the two sections of the overflow conduit, and thus the flow of the fuel to the low-pressure chamber, in order to control fuel injection. Because of the very limited amount of available space in the cylinder head of the engine, the overflow valve is secured on a part of the housing that projects laterally from the pump housing at the level of the cylinder bore, and in such an arrangement the first section of the overflow conduit, that is, the section that can be subjected to injection pressure, cannot lead directly from the pump work chamber through to the overflow valve; it takes an angled course instead. This first section of the overflow conduit therefore comprises a connecting bore, offset laterally with respect to the cylinder bore, and a transverse conduit that connects this connecting bore with a control conduit monitored by the overflow valve. In the unit fuel injector known from the above patent, this transverse conduit leads from the outside from an inflow bore on through the laterally projecting housing part, and finally discharges into the connecting bore. The transverse conduit crosses the control conduit, which in this unit fuel injector comprises a control pressure chamber of the overflow valve, and must be sealed off from the outside, toward the inlet, by a sealing plug. Because of the very high injection pressure, in the range of 1000 bar, such sealing plugs mean there is a constant danger of leakage, or else they are destroyed by the high pressures or are forced out of the bore.